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This is not correct (the stated conclusion implies $H \subset \langle \eta \rangle \cdot \langle \gamma \rangle$, which need not be true), so there must be a typographical error. Probably the conclusion should be $H \cap \Gamma_n \subset \langle \eta \rangle \cdot \langle \gamma \rangle$. Since $\Gamma$ is countable, there are only countably many possibilities for the subset $H$. Given a subset $H_m$ as described, double coset separability implies there is a finite-index normal subgroup $\Gamma_m'$, such that $H_m \cap \Gamma_m' \subset \langle \eta \rangle \cdot \langle \gamma \rangle$. Let $\Gamma_n = \bigcap_{m=1}^n \Gamma_m'$.